Answer to Question #131392 in Differential Equations for Meenu Rani

Question

Consider the Dirichlet problem in an annulus Ω in R3,
Ω={(r,θ) ∈ R2; 0 < a <r <1, −π ≤θ≤π}.
This problem requires one to find a function u(r,θ) in C2(Ω) ∩ C0(¯Ω) such that
∇^2 u(r,θ) = 0, a < r < 1, −π ≤θ ≤π
u(1,θ) = f(θ), −π ≤ θ ≤ π,
u(a,θ) = g(θ), −π ≤ θ ≤ π.
Assume that the solutions can be represented as a superposition of the functions 1,logr, rn cosnθ,rn sinnθ,r−n cosnθ,r−n sinnθ, n = 1,2,..., all of which are harmonic in Ω, and discuss the determination of the coefficients. Assume that f and g are C1 functions.

Answer 1

Dear Meenu Rani, your question requires a lot of work, which neither of our experts is ready to perform for free. We advise you to convert it to a fully qualified order and we will try to help you. Please click the link below to proceed: Submit order